topological degeneracy
{{Use American English|date=January 2019}}{{Short description|Phenomenon in many-body quantum systems
}}
In quantum many-body physics, topological degeneracy is a phenomenon in which the ground state of a gapped many-body Hamiltonian becomes degenerate in the limit of large system size such that the degeneracy cannot be lifted by any local perturbations.{{cite journal | last1=Wen | first1=X. G. |author-link=Xiao-Gang Wen| last2=Niu | first2=Q. | title=Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces | journal=Physical Review B | publisher=American Physical Society (APS) | volume=41 | issue=13 | date=1 April 1990 | issn=0163-1829 | doi=10.1103/physrevb.41.9377 | pmid=9993283 | pages=9377–9396|url=http://dao.mit.edu/~wen/pub/topWN.pdf | bibcode=1990PhRvB..41.9377W}}
Applications
Topological degeneracy can be used to protect qubits which allows topological quantum computation.{{cite journal | last1=Nayak | first1=Chetan | last2=Simon | first2=Steven H. |author-link2=Steven H. Simon| last3=Stern | first3=Ady |author-link3=Ady Stern| last4=Freedman | first4=Michael |author-link4=Michael Freedman| last5=Das Sarma | first5=Sankar |author-link5=Sankar Das Sarma| title=Non-Abelian anyons and topological quantum computation | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=80 | issue=3 | date=2008-09-12 | issn=0034-6861 | doi=10.1103/revmodphys.80.1083 | pages=1083–1159|arxiv=0707.1889 | bibcode=2008RvMP...80.1083N| s2cid=119628297 }} It is believed that topological degeneracy implies topological order (or long-range entanglement {{cite journal | last1=Chen | first1=Xie | last2=Gu | first2=Zheng-Cheng | last3=Wen | first3=Xiao-Gang |author-link3=Xiao-Gang Wen| title=Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order | journal=Physical Review B | volume=82 | issue=15 | date=2010-10-26 | issn=1098-0121 | doi=10.1103/physrevb.82.155138 | page=155138|arxiv=1004.3835 | bibcode=2010PhRvB..82o5138C| s2cid=14593420 }}) in the ground state.{{cite journal | last=Wen | first=X. G. |author-link=Xiao-Gang Wen| title= Topological Orders in Rigid States | journal=International Journal of Modern Physics B | publisher=World Scientific Pub Co Pte Lt | volume=04 | issue=2 | year=1990 | issn=0217-9792 | doi=10.1142/s0217979290000139 | pages=239–271|archive-url=https://web.archive.org/web/20070806075129/http://dao.mit.edu/~wen/pub/topo.pdf|archive-date=2007-08-06|url=http://dao.mit.edu/~wen/pub/topo.pdf | bibcode=1990IJMPB...4..239W}} Many-body states with topological degeneracy are described by topological quantum field theory at low energies.
Background
Topological degeneracy was first introduced to physically define topological order.{{cite journal | last=Wen | first=X. G. |author-link=Xiao-Gang Wen| title=Vacuum degeneracy of chiral spin states in compactified space | journal=Physical Review B | publisher=American Physical Society (APS) | volume=40 | issue=10 | date=1 September 1989 | issn=0163-1829 | doi=10.1103/physrevb.40.7387 | pmid=9991152 | pages=7387–7390| bibcode=1989PhRvB..40.7387W }}
In two-dimensional space, the topological degeneracy depends on the topology of space, and the topological degeneracy on high genus Riemann surfaces encode all information on the quantum dimensions and the fusion algebra of the quasiparticles. In particular, the topological degeneracy on torus is equal to the number of quasiparticles types.
The topological degeneracy also appears in the situation with topological defects (such as vortices, dislocations, holes in 2D sample, ends of a 1D sample, etc.), where the topological degeneracy depends on the number of defects. Braiding those topological defect leads to topologically protected non-Abelian geometric phase, which can be used to perform topologically protected quantum computation.
Topological degeneracy of topological order can be defined on a closed space or an open space with gapped boundaries or gapped domain
walls,{{cite journal | last1=Kitaev | first1=Alexei | last2=Kong | first2=Liang | title=Models for gapped boundaries and domain walls | journal=Commun. Math. Phys. | volume=313 | issue=2 | pages=351–373 | date=July 2012 | issn= 1432-0916 | doi=10.1007/s00220-012-1500-5 |arxiv=1104.5047| bibcode=2012CMaPh.313..351K | s2cid=3070055 }} including both Abelian topological orders {{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | title=Boundary Degeneracy of Topological Order | journal=Physical Review B | volume=91 | issue=12 | date=13 March 2015 | issn= 2469-9969 | doi=10.1103/PhysRevB.91.125124 | page=125124 |arxiv=1212.4863| bibcode=2015PhRvB..91l5124W | s2cid=17803056 }}{{cite journal | last=Kapustin | first=Anton | title=Ground-state degeneracy for abelian anyons in the presence of gapped boundaries | journal=Physical Review B | publisher=American Physical Society (APS) | volume=89 | issue=12 | date=19 March 2014 | issn= 2469-9969 | doi=10.1103/PhysRevB.89.125307 | page=125307 |arxiv=1306.4254 | bibcode=2014PhRvB..89l5307K| s2cid=33537923 }}
and non-Abelian topological orders.
{{cite journal | last1=Lan | first1=Tian | last2=Wang | first2=Juven | last3=Wen | first3=Xiao-Gang | title=Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy | journal=Physical Review Letters | volume=114 | issue=7 | date=18 February 2015 | issn= 1079-7114 | doi=10.1103/PhysRevLett.114.076402 | page=076402 |arxiv=1408.6514 | pmid=25763965 | bibcode=2015PhRvL.114g6402L| s2cid=14662084 }} The application of these types of systems for quantum computation has been proposed.{{cite journal | last1=Bravyi | first1=S. B. | last2=Kitaev | first2=A. Yu. | title=Quantum codes on a lattice with boundary |arxiv=quant-ph/9811052| year=1998 | bibcode=1998quant.ph.11052B }} In certain generalized cases, one can also design the systems with topological interfaces enriched or extended by global or gauge symmetries.{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | last3=Witten | first3=Edward | title=Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions | journal=Physical Review X | volume=8 | issue=3 | date=August 2018 | issn= 2160-3308 | doi=10.1103/PhysRevX.8.031048 | page= 031048 |arxiv=1705.06728| bibcode=2018PhRvX...8c1048W | s2cid=119117766 }}
The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors{{cite journal | last1=Read | first1=N. | last2=Green | first2=Dmitry | title=Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect | journal=Physical Review B | volume=61 | issue=15 | date=15 April 2000 | issn=0163-1829 | doi=10.1103/physrevb.61.10267 | pages=10267–10297|arxiv=cond-mat/9906453 | bibcode=2000PhRvB..6110267R| s2cid=119427877 }}) with trapped defects (such as vortices). In non-interacting fermion systems, there is only one type of topological degeneracy
where number of the degenerate states is given by , where
is the number of the defects (such as the number of vortices).
Such topological degeneracy is referred as "Majorana zero-mode" on the defects.{{cite journal | last=Kitaev | first=A Yu | title=Unpaired Majorana fermions in quantum wires | journal=Physics-Uspekhi | publisher=Uspekhi Fizicheskikh Nauk (UFN) Journal | volume=44 | issue=10S | date=1 September 2001 | issn=1468-4780 | doi=10.1070/1063-7869/44/10s/s29 | pages=131–136|arxiv=cond-mat/0010440 | bibcode=2001PhyU...44..131K| s2cid=9458459 }}
In contrast, there are many types of topological degeneracy for interacting systems.{{cite journal | last=Bombin | first=H. | title=Topological Order with a Twist: Ising Anyons from an Abelian Model | journal=Physical Review Letters | volume=105 | issue=3 | date=14 July 2010 | issn=0031-9007 | doi=10.1103/physrevlett.105.030403 | pmid=20867748 | page=030403|arxiv=1004.1838 | bibcode=2010PhRvL.105c0403B| s2cid=5285193 }}
A systematic description of topological degeneracy is given by tensor category (or monoidal category) theory.